For my final teaching experiment, I selected the “Pass the Problem” formative assessment activity. I designed this task as an introduction to linear inequalities, with the intent that students would make connections with their prior knowledge of inequalities in one variable and linear equations. The content goals of the task were for students to identify differences between linear inequalities and inequalities in one variable, and to construct arguments explaining how they thought about and demonstrated graphing linear inequalities. I formatted the task so that students had about 15 minutes to explore the problems with a partner, 15 minutes to interpret and add to the work of another pair, and 10 minutes to discuss the problems as a class. The “Pass the Problem” task format and my design of the task facilitated the use of several Math Practices. First, because the content was new and students were asked to interpret the work of other students, the task requires students to make sense of problems and persevere in solving them. Many students expressed that they found the task challenging since they had not been explicitly taught the material, but the students were successfully able to use the connections to their prior knowledge of linear equations to make sense of portions of the task. Second, I explicitly asked the students to critique the reasoning of another pair and explain why they agreed or disagreed with the other team’s work (see Appendix A). Students were less successful than I would have liked in this portion of the task, as these are skills we have practiced in the past. I suspect that students were unsuccessful in this area because they were not confident in their own understanding of the problem. Third, students had tools available to them, the strategic use of which (twistable crayons to differentiate their work from the original team’s work, and straight-edges for drawing accurate lines) enhanced the precision of their work. It was particularly important for students to be precise in their work so that other students could successfully interpret their work. Finally, the task required students to identify structure, specifically the connections to their prior knowledge. I launched the task by explaining to students that an important skill in mathematics is the ability to take their prior knowledge and apply it to a new context to “figure out” something new. In this case, connect their knowledge of inequalities in one variable and linear equations to explore graphing linear inequalities. The majority of pairs successfully graphed the lines which corresponded with each linear inequality. They recognized it as slope-intercept form and identified both the slope and y-intercept, and used that information to graph the line. All pairs, even those who did not correctly graph the line, identified the slope and y-intercept on their papers (see Appendices B and C for a sample of student work in this portion of the task) though I did not explicitly prompt them to. One difference between my two implementations of the lesson was the degree to which each class addressed the idea of shading the graph to represent the inequality. In the first class, most groups discussed where the graph would be shaded, but did not show this in their graph. In the second class, slightly less than half of the pairs identified in their work that the inequality would affect the appearance of the graphs. There were several strategies used by students:
During our discussion that concluded each lesson, both classes arrived at the idea of testing multiple points in order to decide where to shade, and I plan to emphasize this idea in our next lesson. Each class also identified that an entire region of the graph would be shaded to reflect the solution set to the inequality. I found it very interesting to further examine each pair’s interpretation of the inequality, and look forward to seeing how the ideas from this lesson carry into their understanding of the material once we explicitly learn it in class. The interpretation of the inequality that I found most interesting, particularly because it was so common in my first implementation of the lesson was that seen in Appendices E and F, that the slope of the line determined the direction of shading, rather than the symbol of the inequality. Overall, I was impressed with my students’ work on the task and the connections that they made to their prior knowledge, even if their work was not technically correct. One modification I made from my original implementation of a “Pass the Problem” task was to provide less structure (eg. not specifying for students to explicitly identify the slope and y-intercept), in the hopes of increasing the cognitive demand of the task. I think that the cognitive demand of this task was already fairly high, as it required use of different representations, connections to prior knowledge, and interpretation of other students’ work. That said, I do not think that removing the structure of requiring students to explicitly identify slope and y-intercept made the task overly difficult. I did provide some scaffolding for the students in the form of sub-questions that had the purpose of focusing their attention on key ideas. I think that with the demand of this task, these questions were integral in keeping the students engaged and focused on the task, rather than being overwhelmed. The addition of the Team Two Analysis (see Appendix I) was a modification I made since my first implementation of a “Pass the Problem” activity. One observation that I have made with these students is that if I do not explicitly direct students to do something, they probably will not do it. As a result, I was concerned that when students switched their work with another pair, they would not try to understand the other team’s work if it differed from their own. I think that adding a formal structure for students to use when examining the other team’s work was beneficial for this reason, but I also observed that students were not particularly successful at trying to explain what the other team might have been thinking or why they did the work as they did. This is a skill that the students have used successfully in the past, so I suspect that this struggle resulted from the students being unconfident in justifying or explaining their own work, let alone trying to explain someone else’s. That said, I think that the structure organizing how they looked at each other’s work helped the students to have a better understanding of the problem. I have made this conclusion because I overheard several conversations of pairs discussing that they felt like the work they were examining was more correct than their own. Even if the new work was not completely correct, these conversations show me that the students were thinking about the ideas and their meaning. I made two other minor modifications since my first implementation that I believe were successful in this enactment of the “Pass the Problem” formative assessment activity. First, I had all pairs working on the same problem. I made this modification so that after switching papers, students were looking at the same problem that they had worked on in the first portion of the task. Because this task was introducing new material, I concluded that this would positively impact student thinking, because the context of the other team’s work would be familiar but that the work would likely be different because the students had not yet learned the correct way to complete the problems. In this way, the cognitive demand was not lowered, but the students were also not overwhelmed by trying to figure out a new problem while also trying to interpret someone else’s work. My other modification was to switch their papers with pairs from other tables, rather than the pair across from them. One reason for this was having an odd number of pairs in the class. My second (and primary) reason was that last time I found that I had difficulties preventing crosstalk at the table, so after switching papers, the original team explained their work to the second team, rather than the second team needing to interpret the original team’s work. With this shift, it forced the students to actually interpret their work, rather than simply asking them to explain it. Beyond these changes, I tried to keep the structure fairly similar due to the overall success I found in my previous implementation. I think that a great deal of complex thinking was generated in this activity, and that the students will have a stronger understanding of this content as a result.
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